Evanescent Effects Can Alter Ultraviolet Divergences
in Quantum Gravity without Physical Consequences
Evanescent operators such as the Gauss-Bonnet term have vanishing perturbative matrix elements in exactly dimensions. Similarly, evanescent fields do not propagate in ; a three-form field is in this class, since it is dual to a cosmological-constant contribution. In this Letter, we show that evanescent operators and fields modify the leading ultraviolet divergence in pure gravity. To analyze the divergence, we compute the two-loop identical-helicity four-graviton amplitude and determine the coefficient of the associated (non-evanescent) counterterm studied long ago by Goroff and Sagnotti. We compare two pairs of theories that are dual in : gravity coupled to nothing or to three-form matter, and gravity coupled to zero-form or to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the one-loop conformal anomaly — the coefficient of the Gauss-Bonnet operator — changes under -form duality transformations. We concur, and also find that the leading divergence changes under duality transformations. Nevertheless, in both cases the physical renormalized two-loop identical-helicity four-graviton amplitude can be chosen to respect duality. In particular, its renormalization-scale dependence is unaltered.
pacs:04.65.+e, 11.15.Bt, 11.25.Db, 12.60.Jv
UCLA/15/TEP/101 SLAC–PUB–16332 CALT-2015-036 CERN-PH-TH-2015-162
Although theories of quantum gravity have been studied for many decades, basic questions about their ultraviolet (UV) structure persist. One subtlety is the conformal anomaly111Einstein gravity is not conformally invariant, so this is not an anomaly in the traditional sense., also known as the Weyl or trace anomaly ConformalAnomaly (). At one loop, the conformal anomaly provides the coefficient of the Gauss-Bonnet (GB) term. The physical significance of this relationship has not been settled, however. In particular, Duff and van Nieuwenhuizen showed that the conformal anomaly changes under duality transformations of -form fields, suggesting that theories related through such transformations are quantum-mechanically inequivalent DuffInequivalence (). In response, Siegel argued that this effect is a gauge artifact and therefore not physical SiegelEquivalence (); Fradkin and Tseytlin and Grisaru et al. have also argued that duality should hold at the quantum level FradkinTseytlinEquivalence (). Furthermore, for external states, one-loop divergences in gravity theories coupled to two-form antisymmetric tensors are unchanged under a duality transformation relating two-forms to zero-form scalars OneLoopEquivalence (). However, as we shall see, intuition based on one-loop analyses can be deceptive.
As established in the seminal work of ’t Hooft and Veltman HooftVeltman (), pure gravity is finite at one loop because the only available counterterm is the GB term, which integrates to zero in a topologically trivial background. While amplitudes with external matter fields diverge at one loop, amplitudes with only external gravitons remain finite. At two loops, however, pure gravity diverges, as demonstrated explicitly by Goroff and Sagnotti GoroffSagnotti () and confirmed by van de Ven vandeVen ().
In this Letter, we investigate the UV properties of the two-loop amplitude for scattering of four identical-helicity gravitons, including the effect of -form duality transformations. We use dimensional regularization, which forces us to consider the effects of evanescent operators like the GB term, which are legitimate operators in dimensions but vanish (or are total derivatives) in four dimensions. We show that the GB counterterm is required to cancel subdivergences and reproduce the two-loop counterterm coefficient found previously GoroffSagnotti (); vandeVen ().
Evanescent operators are well-studied in gauge theory (see e.g. Ref. Evanescent ()), where they can modify subleading corrections. In contrast, we find that evanescent effects can alter the leading UV divergence in gravity.222Effects of the GB term have also been studied in renormalizable, but non-unitary, gravity R2GB (). Despite this change in the UV divergence, the physical dependence of the renormalized amplitude on the renormalization scale remains unchanged. This break in the link between the UV divergence and the renormalization-scale dependence is unlike familiar one-loop examples. We arrive at a similar conclusion when comparing the divergences and renormalization-scale dependences in gravity coupled to scalars versus antisymmetric-tensor fields.
Pure gravity is defined by the Einstein-Hilbert Lagrangian,
where and the metric signature is . We also augment by matter Lagrangians for one of the following: scalars, two-form fields (antisymmetric tensors) or three-form fields:
Here is a scalar field and and are the field-strengths of the two- and three-form antisymmetric-tensor fields and . The index labels distinct fields. Standard gauge-fixing for the two- and three-form actions, as well as for , leads to a nontrivial ghost structure. We avoid such complications by using the generalized unitarity method UnitarityMethod (); DDimensionalUnitarity (); AllPlusQCD (), which directly imposes appropriate -dimensional physical-state projectors on the on-shell states crossing unitarity cuts.
Under a duality transformation, in four dimensions the two-form field is equivalent to a scalar:
and the three-form field is equivalent to a cosmological-constant contribution via
As usual, we expand the graviton field around a flat-space background: . Similarly, we expand the scalar, two-form field and three-form field around trivial background values. It is interesting to note that the three-form field has been proposed as a means for neutralizing the cosmological constant Cosmology ().
which is proportional to the conformal anomaly. The calculations of the conformal anomaly and of the UV divergence are essentially the same, except that we replace a graviton polarization tensor with a trace over indices. Contracting Eq. (5) with four on-shell graviton polarization tensors gives zero. This is because the GB combination is evanescent in : It is a total derivative and vanishes when integrated over a topologically trivial space; hence pure Einstein gravity is finite at one loop HooftVeltman (). In a topologically nontrivial space, the integral over the GB term gives the Euler characteristic. When matter is added to the theory, the four-graviton amplitude is still UV finite at one loop, although divergences appear in amplitudes with external matter states.
Using the unitarity method, we verified Eq. (5) by considering the one-loop four-graviton amplitude with external states in arbitrary dimensions and internal ones in dimensions. On-shell scattering amplitudes are sensitive only to the coefficient of the operator, because the and operators can be eliminated by field redefinitions at leading order in the derivative expansion. The GB combination is especially simple to work with in dimensional regularization since there are no propagator corrections in any dimension Zwiebach ().
For the case of antisymmetric tensors coupled to gravity, another relevant one-loop four-point divergence is that of two gravitons and two antisymmetric tensors, generated by the operator,
Like the GB term, this operator is evanescent. In particular, in , we can dualize the antisymmetric tensors to scalars, which collapses the Riemann tensor into the Ricci scalar and tensor. Under field redefinitions, they can be eliminated in favor of the dualized scalars, removing the one-loop divergence in two-graviton two-antisymmetric-tensor amplitudes with external states. The four-scalar amplitude does diverge.
The change in Eq. (5) under duality transformations is central to the claim by Duff and van Nieuwenhuizen of quantum inequivalence under such transformations DuffInequivalence (). Here we analyze their effects on the two-loop amplitude. First let us note that our unitarity-based evaluation of Eq. (5) sews together physical, gauge-invariant tree amplitudes. This explicitly demonstrates that the numerical coefficient of the term in Eq. (5) is gauge invariant, in contrast to implications of Ref. SiegelEquivalence (). This gauge invariance suggests that by two loops, Eq. (5) could lead to duality-violating contributions to non-evanescent operators. To see if this happens, we must account for subdivergences and renormalization.
At two loops, pure gravity diverges in . The coefficient of this divergence was determined by Goroff and Sagnotti GoroffSagnotti () from a three-point computation in the standard regularization scheme and later confirmed by van de Ven vandeVen ():
In order to reproduce the Goroff and Sagnotti result, we evaluate the identical-helicity four-graviton amplitude. This is the simplest amplitude containing the two-loop divergence (7). While a four-point amplitude may seem to be unnecessarily complicated with respect to a three-point function, there are several advantages to considering an amplitude for a physical process with real momenta. The first is that we can use the unitarity method to obtain a compact integrand UnitarityMethod (). This method is particularly efficient for identical-helicity particles, having been used to obtain compact integrands for the gauge-theory case AllPlusQCD (). More importantly, the question of quantum equivalence under duality transformations can only be properly answered in the context of physical observables, such as renormalized and infrared-subtracted scattering amplitudes entering physical cross-sections.
To facilitate comparisons to the two-loop four-point amplitude, we need the divergence (7) inserted into the four-plus-helicity tree amplitude:
and , and are the usual Mandelstam invariants. The last factor is a pure phase constructed from the spinor products and defined in, for example, Ref. Review ().
Fig. 1 shows that there are three types of contributions to consider: (a) the bare two-loop contribution, (b) the one-loop single-counterterm subtraction and (c) the double-counterterm subtraction. One might expect the net subdivergence subtractions (b) and (c) each to be zero because there are no corresponding one-loop divergences. However, this is not correct. A careful analysis of the two-loop integrands LongPaper () reveals subdivergences associated with the GB term (5). For the case of two-forms, a subdivergence corresponding to in Eq. (6) must also be subtracted. In principle, when three-forms are present, there might have been subdivergences due to operators containing three-forms, but these do not appear. It is somewhat surprising that there are subdivergences at two loops without any corresponding one-loop divergences in . However, because some legs external to the subdivergence are in dimensions, the cancellations that are specific to do not occur.
While Goroff and Sagnotti also subtracted subdivergences, they did so integral by integral, rather than tracking the operator origin of the subdivergences as we do. Here we use dimensional regularization for both infrared and UV divergences; we subtract the well-known infrared singularities IRPapers () from the final result.
We evaluate the bare and single-subtraction contributions via the unitarity method. We take the external legs to be identical-helicity gravitons and each internal leg to be -dimensional. The bare integrand obtained in this way is similar to integrands found earlier for gauge theory DDimensionalUnitarity (); AllPlusQCD () and for the “double-copy” theory containing a graviton, an antisymmetric tensor and a dilaton DoubleCopyGravity (). A key property of these integrands is that they vanish when the loop momenta are taken to reside in , yet the amplitudes are still nonvanishing. This phenomenon is related to the observation by Bardeen and Cangemi AllPlusAnomaly () that the nonvanishing of identical-helicity amplitudes is connected to an anomaly in the self-dual sector.
We follow the same regularization prescriptions used in Ref. AllPlusQCD (), where algebraic manipulations on the integrand are performed with . We use the ’t Hooft-Veltman variant: We place the external momenta and polarizations in and take the loop momenta and internal states to reside in dimensions. Here we focus on the UV divergences and defer presentation of the integrands and finite terms in the amplitudes to Ref. LongPaper ().
We integrate over the loop momenta with the same techniques used to obtain two-loop four-point helicity amplitudes in QCD, including their finite parts TwoLoopQCDIntegration (). As a cross check, we also directly extract the UV divergences using masses to regulate the infrared DoubleCopyGravity ().
Consider first the case of three-forms coupled to gravity. In Table 1, we give both the divergence and renormalization-scale dependence of each of the three components illustrated in Fig. 1. In the bare and one-loop single-insertion components, the dependence, where is the renormalization scale, is proportional to the UV divergence. For the bare two-loop part, the coefficient is twice the coefficient of the divergence. For the single counterterm, it is equal to the coefficient, and for the double-insertion tree contribution, it vanishes. This follows from dimensional analysis of the loop integrals, with measure per loop, requiring an overall factor of at loops. The counterterm subtractions are pure poles that do not carry such factors. In the sum over terms, there is no simple relation between the and the coefficients, in contrast to many textbook examples at one loop.
As seen from the last line of Table 1, with no three-form fields we match exactly the Goroff and Sagnotti divergence (8). The addition of three-form fields shifts the divergence from the pure gravity result. One might think that this shift would lead to a physical change in the scattering amplitudes through a different dependence on . However, the column of Table 1 shows that the -dependence of the bare and single-counterterm contributions precisely cancels in the sum. The scale dependence is therefore unaffected by three-form fields. The differences in the divergent parts can be removed by adjusting the coefficient of the counterterm. We have also obtained the amplitude’s finite parts LongPaper (). Their form allows for a finite subtraction that completely eliminates the effects of three-form fields in the two-loop renormalized identical-helicity amplitude.
We now turn to the case of duality transformations between antisymmetric-tensor fields and scalars. In Tables 2 and 3, the coefficients of and terms are collected. The tables show that, while the individual components are quite different and the final divergence changes under duality transformations, scalars and two-forms have exactly the same renormalization-scale dependence. As for the case of three-forms, we find that the UV divergence does depend on the field representations, but the renormalization-scale dependence does not. Again, finite subtractions can be found to make the dual pair of renormalized amplitudes identical LongPaper ().
where () is the number of bosonic (fermionic) four-dimensional states in the theory. We only computed Eq. (10) explicitly for , but the identical-helicity graviton amplitude vanishes in supersymmetric theories, forcing Eq. (10) to be proportional to .
The dependence is clearly a more appropriate quantity for deciding whether a theory should be thought of as nonrenormalizable. If the coefficient of the is nonvanishing, as is the case for pure gravity, the coefficient will run, and we consider such a theory to be nonrenormalizable. Our result shows that instead of focusing on the divergences, one should study the coefficient to see if there is a principle that can be applied to set it to zero. One obvious useful principle is that renormalization schemes should be chosen that maintain the equality of theories related by duality transformations.
In this light, one might wonder if the recently-computed four-loop divergence of pure supergravity N4FourLoop () is an artifact of the particular formulation of the theory that was used. However, with the uniform mass infrared regulator used in that calculation, extensive checks reveal that all subdivergences cancel. Therefore the coefficient of is proportional to that of the divergence. When matter multiplets are added there are one-loop subdivergences, but those are not evanescent. In other formulations, it is possible that the divergences will change, but we do not expect the coefficients to change.
In summary, our investigation of the ultraviolet divergences of nonsupersymmetric gravity reveals a number of striking features. The first is the nontrivial role of the conformal anomaly and the associated evanescent Gauss-Bonnet term entering subdivergences. It is remarkable that a term that vanishes in four dimensions can contribute directly to the leading divergence of a graviton amplitude. Another important feature is that the integrand of the identical-helicity amplitude vanishes if the loop momenta are taken to be four-dimensional; this feature of identical-helicity amplitudes, which follows straightforwardly from unitarity, is also tied to anomalous behavior AllPlusAnomaly (). Similar connections to anomalous behavior MarcusAnomaly () were noted in the four-loop divergence of pure supergravity N4FourLoop ().
A key lesson is that under duality transformations, the values of two-loop divergences can change, contrary to the situation at one loop OneLoopEquivalence (). However, the difference in these divergences are unphysical, in the sense that they can be absorbed into a redefinition of the coefficient of a local operator. In other words, our results for scattering amplitudes are consistent with quantum equivalence under duality transformations when that equivalence allows for the adjustment of coefficients of higher-dimension operators. The dependence on the renormalization scale does not change under duality transformations in the examples we studied; it is a more appropriate measure of the UV properties of the theory. It would be quite interesting to establish this property beyond two loops. Together with recent examples of ultraviolet finiteness in supergravity amplitudes, despite the existence of seemingly valid counterterms N4gravThreeLoops (); N5FourLoop (), the results summarized in this Letter show that much more remains to be learned about both duality at the quantum level and the ultraviolet structure of gravity theories.
We especially thank David Kosower, Radu Roiban, Augusto Sagnotti and Raman Sundrum for many useful and interesting discussions and suggestions. We also thank Luis Alvarez-Gaume, John Joseph Carrasco, Stanley Deser, Paolo Di Vecchia, Gary Horowitz, Henrik Johansson, Tim Jones, Kelly Stelle and Mark Wise for helpful discussions. This material is based upon work supported by the Department of Energy under Award Number DE-SC0009937 and contract DE–AC02–76SF00515 and the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics. CC is supported by a DOE Early Career Award DE-SC0010255 and a Sloan Research Fellowship. ZB and LD are grateful to the Simons Foundation for support and to the Walter Burke Institute at Caltech for hospitality. CC and LD thank the Aspen Center for Physics and the NSF Grant #1066293 for hospitality. SD and JN gratefully acknowledge Mani Bhaumik for his generous support.
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